A regularity result for minimal configurations of a free interface problem

https://doi.org/10.1007/s40574-021-00285-6

A regularity result for minimal configurations of a free interface problem

Lorenzo Lamberti¹

Received: 9 November 2020 / Accepted: 3 April 2021 / Published online: 19 April 2021

© The Author(s) 2021

Abstract

We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open regionΩ ⊆ Rⁿ. We will deal with the energy functionalF(v,E) := ´

Ω[F(∇v)+1_EG(∇v)+ f_E(x, v)]d x + P(E, Ω). The bulk energy depends on a functionvand its gradient∇v. It consists in two strongly quasi-convex functionsFandG, which have polinomialp-growth and are linked with their p-recession functions by a proximity condition, and a function fE, whose absolute value satisfies aq-growth condition from above. The surface penalization term is proportional to the perimeter of a subsetEinΩ. The termf_Eis allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated withF. The same condition turns out to be crucial in the proof of the regularity result as well. If(u,A)is a minimal configuration, we prove thatuis locally Hölder continuous andAis equivalent to an open setA. We finally get˜ P(A, Ω)=Hⁿ⁻¹(∂A˜∩Ω).

Keywords Free boundary problem·Perimeter penalization·Regularity·Nonlinear variational problem

Mathematics Subject Classification 49N60·49Q20

1 Introduction and statement

The problem of finding the minimal energy configuration of a mixture of two materials in a bounded open setΩ⊆Rⁿ, penalized by the perimeter of the contact interface between the two materials, has been fully examined in mathematical literature (see for example [2,3,6,8, 10,15,17–20]).

Let p > 1 and define A(Ω) as the set of all subsets of Ω with finite perimeter.

ConsiderF,G∈C¹(Rⁿ)and define fE:=g+1Eh, whereE∈A(Ω)andg,h:Ω×R→ Rare two Borel measurable and lower semicontinuous functions with respect to the real vari-

B

1 Dipartimento di Matematica, Universitá degli Studi di Salerno, Fisciano, Italy

able. We will deal with the following energy functional:

F(v,E):=

ˆ

Ω[F(∇v)+1EG(∇v)+ fE(x, v)]d x+P(E, Ω), where(v,E)∈

u₀+W₀^1,p(Ω)

×A(Ω), withu₀∈W^1,p(Ω). The regularity of minimizers (u,A)of the functionalFwas recently investigated in [6,9,10] for the constrained problem where the volume of the region AinΩ is prescribed but the forcing term f_Ais zero. In the quadratic case p=2 Ambrosio and Buttazzo [3] proved the regularity for minimizers ofF in the case that fAis not zero. We are going to extend this result to functionals with polinomial growth.

We assume that there exist some positive constantsl,L, α, βandμ≥0 such that – FandGhavep-growth:

0≤F(ξ)≤L(μ²+ |ξ|²)^p2, (F1) 0≤G(ξ)≤βL(μ²+ |ξ|²)^p2, (G1) for allξ∈Rⁿ.

– FandGare strongly quasi-convex:

ˆ

ΩF(ξ+ ∇ϕ)d x≥ ˆ

Ω[F(ξ)+l(μ²+ |ξ|²+ |∇ϕ|²)^p−22 |∇ϕ|²]d x, (F2) ˆ

ΩG(ξ+ ∇ϕ)d x≥ ˆ

Ω[G(ξ)+αl(μ²+ |ξ|²+ |∇ϕ|²)^p−22 |∇ϕ|²]d x, (G2) for allξ ∈Rⁿandϕ∈C_c¹(Ω).

– there exist two positive constantst0,aand 0<m < psuch that for everyt > t0and ξ ∈Rⁿwith|ξ| =1, it holds

Fp(ξ)− F(tξ) t^p

≤ a

t^m, (F3)

Gp(ξ)−G(tξ) t^p

≤ a

t^m, (G3)

whereFpandGpare thep-recession functions ofFandG(see Definition2.1).

We remark that the proximity conditions (F3) and (G3) are trivially satisfied ifFandGare positivelyp-homogeneous.

The first of the following assumptions ongandhis essential to prove the existence of a minimal configuration. The same condition turns out to be crucial in the proof of the regularity result as well. We assume that there exist a functionγ ∈L¹(Ω)and two constantsC₀ >0 andk ∈R, withk < ₂p−1^l λ, beingλ=λ(Ω)the first eigenvalue of the p-Laplacian onΩ with boundary datumu0, such that

– gandhsatisfy the following assumptions:

g(x,s)≥γ (x)−k|s|^p, h(x,s)≥γ (x)−k|s|^p, (1.1) for almost all(x,s)∈Ω×R.

– gandhsatisfy the following growth conditions:

|g(x,s)| ≤C₀(1+ |s|^q), |h(x,s)| ≤C₀(1+ |s|^q), (1.2)

for all(x,s)∈Ω×R, with the exponent q∈

[p,+∞) ifn=2, [p,p^∗) ifn>2 fixed.

We want to study the following problem:

min

(v,E)∈

u0+W₀^1,p(Ω)

×A(Ω)F(v,E). (P) The main result of the paper is the following theorem about the regularity of solutions of problem (P).

Theorem 1.1 Let(u,A)be a solution of (P). Then 1. u is locally Hölder continuous.

2. A is equivalent to an open setA, that is˜

Lⁿ(AA)˜ =0 and P(A, Ω)= P(A, Ω)˜ =Hⁿ⁻¹(∂A˜∩Ω).

The idea of its proof is similar to that of Theorem 2.2 in [3], which in turns relies on the ideas introduced in [7]. The regularity ofuis proved in Theorem4.1and the regularity ofA follows from Proposition5.1. The proof will be discussed in the final section.

The same arguments can be used to treat also the volume-constraint problem min

(v,E)∈

u0+W₀^1,p(Ω)

×A(Ω) Lⁿ(E)=d

F(v,E), (Q)

for some 0<d<Lⁿ(Ω). The following theorem holds true.

Theorem 1.2 There existsλ0>0such that if(u,A)is a minimizer of the functional F_λ(v,E)=

ˆ

Ω[F(∇v)+1EG(∇v)+ fE(x, v)]d x+P(E, Ω)+λ|Lⁿ(E)−d|

for someλ ≥λ0 and among all configurations(v,E)such thatv ∈u0+W₀^1,p(Ω)and E ∈ A(Ω), thenLⁿ(A) = d and(u,A)is a minimizer of problem(Q). Conversely, if (u,A)is a minimizer of the problem(Q), then it is a minimizer ofF_λ, for allλ≥λ0. The proof of the previous theorem is a straightforward adaptation of the proof of Theorem 1.4 in [6]. The term concerning the function fE can be treated as a constant, thanks to the boundedness stated in Theorem4.1. We finally remark that the termλ|Lⁿ(E)−d|in the functionalF_λcan be inglobed in f_E, since it is bounded. For this reason, Theorem1.1is still valid also for minimal configurations of_Fλand, consequently, for solutions of problem (Q).

2 Notation and preliminary results

Throughout the paper we denote by·,·and·respectively the Euclidean inner product inRⁿ and the associated norm. We writeLⁿ for the Lebesgue measure. Furthermore, we denote byB_r(x)the ball centered inx ∈Rⁿ with radiusr>0 (ifx =0, we write simply

B_r), byωnthe measure ofB₁, and withQ_r(x)the cube centered inx∈Rⁿwith sider>0.

We write the symbolsand→referring to weak and strong convergence, respectively.

We often denote byca general constant that could vary from line to line, even within the same line of estimates. Relevant dependencies on parameters and special constants will be suitably emphasized using brackets.

Throughout this section we denote with H a function belonging toC¹(Rⁿ)and satisfying for some positive constantsl˜andL˜the same kind of assumptions imposed onFandG:

0≤H(ξ)≤ ˜L(μ²+ |ξ|²)^p2, ˆ

ΩH(ξ+ ∇ϕ)d x≥ ˆ

Ω[H(ξ)+ ˜l(μ²+ |ξ|²+ |∇ϕ|²)^p−22 |∇ϕ|²]d x,

for allξ ∈Rⁿandϕ∈C¹_c(Ω). We collect some definitions and well-known results that will be used later. We start giving the definition ofp-recession function ofH.

Definition 2.1 The p-recession function ofHis defined by H_p(ξ):=lim sup

t→+∞

H(tξ) t^p , for allξ∈Rⁿ.

Remark 2.2 It’s clear thatHpis positively p-homogeneous, which means that H_p(sξ)=s^pH(ξ),

for allξ∈Rⁿands>0. It’s also true that the growth condition ofHimplies the following growth condition ofH_p:

0≤Hp(ξ)≤ ˜L|ξ|^p, for anyξ∈Rⁿ.

Next lemma establishes strong quasi-convexity ofHp, provided H verifies an appropriate growth condition. Its proof is in [12] (Lemma 2.8).

Lemma 2.3 Let H as above. If there exist two positive constantst˜₀,d and˜ 0<m˜ < p such that for every t>t˜₀andξ ∈Rⁿwith|ξ| =1, it holds

H_p(ξ)− H(tξ) t^p

≤ d˜ t^m˜,

then ˆ

ΩHp(ξ+ ∇ϕ)d x≥ ˆ

Ω[Hp(ξ)+ ˜l(|ξ|²+ |∇ϕ|²)^p−22 |∇ϕ|²]d x, for allξ ∈Rⁿandϕ∈C_c¹(Ω).

Let’s recall some other useful lemmas.

Lemma 2.4 Let H be as above. It holds that

|∇H(ξ)| ≤2^pL(μ˜ ²+ |ξ|²)^p−12 , for allξ ∈Rⁿ.

Lemma 2.5 Let H as above. There exists a positive constantc˜= ˜c(p,l˜,L˜, μ)such that H(ξ)≥ l˜

2(μ²+ |ξ|²)^p2 − ˜c, for allξ ∈Rⁿ.

The proof of Lemma2.4can be found in [14] (Lemma 5.2), while Lemma2.5is proved in [6] (Lemma 2.3). We define the auxiliary function

V(ξ)=(μ²+ |ξ|²)^p−24 ξ,

for allξ ∈ Rⁿ. Next Lemma has been proved in [13] (Lemma 2.1) for p ≥2 and in [1]

(Lemma 2.1) for 1<p<2.

Lemma 2.6 There exists a constant c=c(n,p)such that 1

c(μ²+ |ξ|²+ |η|²)^p−22 ≤|V(ξ)−V(η)|²

|ξ−η|² ≤c(μ²+ |ξ|²+ |η|²)^p−22 , for allξ, η∈Rⁿ.

Lemma 2.7 Let{u_h}h∈N ⊆ W^1,p(B₁)and u ∈ W^1,p(B₁)such that u_hu in W^1,p(B₁). Assume that{∇u_h}h∈Nis bounded in L^p(B₁). If

h→+∞lim ˆ

B1

ψ|V(∇uh)−V(∇u)|²d y=0, ∀ψ∈C_c^∞(B1) s.t. 0≤ψ ≤1, then u_h→u in W_loc^1,p(B₁).

The proof of the previous lemma follows from Lemma2.6. If p ≥ 2 the hypothesis of boundedness of{∇uh}h∈Nis superfluous. If 1< p <2, by Hölder inequality we gain the stated result.

The following theorem has been proved in [12] (Theorem 2.2).

Theorem 2.8 Let H be as above and letv∈W^1,p(Ω)be a local minimizer of the functional H(w, Ω)=

ˆ

ΩH(∇w)d x,

wherew ∈ v+W₀^1,p(Ω). Thenvis locally Lipschitz-continuous inΩand there exists a constant c=c(n,p,l˜,L˜) >0such that

ess sup_BR

2(x0)(μ²+ |∇v|²)^p2 ≤c

BR(x0)(μ²+ |∇v|²)^2pd y, for all BR(x0)⊆Ω.

Corollary 2.9 Let_H andv ∈W^1,p(Ω)be as in Theorem2.8. Then there exists a constant c_H =c_H(n,p,l,˜ L) >˜ 0such that

ˆ

B_r(x₀)(μ²+ |∇v|²)^2pd y≤c_H r

R _nˆ

B_R(x₀)(μ²+ |∇v|²)^2pd y, for all B_R(x₀)⊆Ωand0<r<R.

3 Existence of minimizing couples

Theorem 3.1 The minimum problem(P)admits at least a solution.

Proof We initially remark that problem (P) can be written as follows:

E∈Amin(Ω){E(E)+P(E, Ω)}, (3.1) where

E(E)= min

v∈u0+W₀^1,p(Ω)

ˆ

Ω[F(∇v)+1EG(∇v)+ fE(x, v)]d x (3.2) SinceF,Gare strongly quasi-convex andg,hare lower semicontinuous in the real variable s, the functionalFis lower semicontinuous with respect to the weak convergence of∇vhin L^pand the strong converge ofvhinL^p(see [5] or [16]). Moreover, the coerciveness of

ˆ

Ω[F(∇v)+1_EG(∇v)]d x

is granted by Lemma 2.5. Therefore the minimum problem (3.2) admits a solution. Let {A_h}h∈N⊆A(Ω)be a minimizing sequence for problem (3.1). It follows that the sequence {P(Ah, Ω)}h∈N is bounded and so, by compactness, there exists A ∈ A(Ω) such that 1A_h → 1Ain L¹_loc(Ω). Letuh ∈ u0 +W₀^1,p(Ω)a solution of problem (3.2) associated withAh, for allh∈N. The sequence{uh}_h∈Nis bounded inW^1,p(Ω); indeed, by (1.1) and Poincaré inequality we obtain

min

v∈u₀+W₀^1,p(Ω)F(v, Ω)≥F(u_h,A_h)≥l ˆ

Ω|∇u_h|^pd x+ ˆ

Ωγd x−k ˆ

Ω|u_h|^pd x

≥l ˆ

Ω|∇uh|^pd x+ ˆ

Ωγd x−2^p−1k

× ˆ

Ω|uh−u0|^pd x−2^p−1k ˆ

Ω|u0|^pd x

≥(l−2^p−1kλ) ˆ

Ω|∇uh|^pd x+ ˆ

Ωγd x−2^p−1k ˆ

Ω|u0|^pd x.

Hence, we can extract a subsequence (not relabelled) such thatu_hu in W^1,p(Ω). By definition of minimum we infer

E(A)≤ ˆ

Ω[F(∇u)+1_AG(∇u)+ f_A(x,u)]d x.

Applying again Ioffe lower semicontinuity result (see for instance [16] or [4], Theorem 5.8) to the integrand

(x,s1,s2, ξ):=F(ξ)+s1G(ξ)+g(x,s2)+s1h(x,s2), wherex ∈Ω,s₁∈ [0,1],s₂∈Randξ∈Rⁿ, we obtain

E(A)≤ ˆ

Ω[F(∇u)+1AG(∇u)+ fA(x,u)]d x= ˆ

Ω(x,1A,u,∇u)d x

≤lim inf

h→+∞

ˆ

Ω(x,1A_h,uh,∇uh)d x=lim inf

h→+∞E(Ah).

Therefore, by the lower semicontinuity of perimeter we finally gain E(A)+P(A, Ω)≤lim inf

h→+∞[E(A_h)+P(A_h, Ω)],

which proves thatAis a minimizer of problem (3.1) and so(u,A)is a minimizing couple of

problem (P).

4 Higher integrability and Hölder continuity of minimizers

The following theorem shows that local minimizers of the functionalF(·,E), withE ∈ A(Ω)fixed, are Hölder continuous and a higher integrability property for the gradient holds true. The proof of this result is standard and can be carried on adopting the obvious adaptation in the proof of Theorem 3.1 in [3].

Theorem 4.1 Let(u,A)be a solution of (P). Then the following facts hold:

– u is locally bounded inΩ by a constant depending only on n,p,q, α, β,l,L, μ,C₀, uL^p(Ω)and is locally Hölder continuous inΩ.

– LetΩ0Ω,τ =dist(Ω0, ∂Ω)and K = {x∈Ω : dist(x, Ω0)≤^τ₂}. Then there exist two constantsγ > 0and r > p depending only on n,p,q, β,l,L, μ,C0,uL^∞(K)

such that ˆ

QR

2(y)|∇u|^rd x≤γ

Rⁿ

1−^r_p ˆ

QR(y)|∇u|^pd x ^r

p +Rⁿ

,

for all y∈Ω0and QR(y)⊆K .

5 Regularity of the set

The following proposition is the main result of this section and also the main ingredient to prove Theorem1.1.

Proposition 5.1 Let(u,A)be a solution of (P). Then for every compact set K ⊆ Ωthere exists a constantξ∈(0,dist(K, ∂Ω))such that if y∈K and for someρ < ξit holds

ˆ

B_ρ(y)[F_p(∇u)+1_AG_p(∇u)]d x+P(A,B_ρ(y)) < ξρⁿ⁻¹, then

η→0limη¹⁻ⁿ ˆ

B_η(y)[F_p(∇u)+1_AG_p(∇u)]d x+P(A,B_η(y))

=0.

The proof of the previous proposition relies on Proposition5.5, which is an iteration of the decay estimate in Theorem5.4. The following definition is crucial in the rescaling argument used in the proof of Theorem5.4(see (5.11)).

Definition 5.2 (Asymptotically minimizing sequence) Let {(uh,Ah)}h∈N ⊆ W^1,p(B1)× A(B1)and{λh}h∈N ⊆ R⁺. We say that the sequence{(uh,Ah)}h∈Nisλh-asymptotically minimizing if and only if for any compact set K ⊆ B₁ and any couple {(u_h,A_h)} ⊆

W^1,p(B₁)×A(B₁)formed by a bounded sequence{u_h}n∈NinW^1,p(B₁)with spt(u_h−u_h)⊆ K and a sequence of sets{A_h}n∈NwithA_hA_h ⊆K, we have

ˆ

B₁[F_p(∇u_h)+1_AhG_p(∇u_h)]d y+λhP(A_h,B)

≤ ˆ

B1

[Fp(∇u_h)+1_A

hGp(∇u_h)]d y+λhP(A_h,B)+ηh,

(5.1)

where{ηh}h∈N⊆Ris an infinitesimal sequence.

In the proof of Theorem 5.4 we will show that the sequence of appropriately rescaled minimal configurations of problem (P) is asymptotically minimizing. The following theorem is concerned with the behaviour of asymptotically minimizing sequences.

Theorem 5.3 Let{λh}h∈N ⊆ R⁺and{(uh,Ah)}h∈N ⊆ W^1,p(B1)×A(B1). Assume that (uh,A_h)isλh-asymptotically minimizing and that

(i) ´

B1[Fp(∇uh)+1A_hGp(∇uh)]d y+λhP(Ah,B1)

h∈Nis bounded.

(ii) uhu in W^1,p(B1).

(iii) 1A_h →1Ain L¹(B1)andλh → +∞.

(iv) G_p(∇uh)is locally equi-integrable in B₁. Then

(a) uh →u in W_loc^1,p(B1).

(b) λhP(Ah,B_ρ)→0, for allρ∈(0,1).

(c) A= ∅or A=B₁and u minimizes the functional´

B₁[Fp(∇v)+1_AG_p(∇v)]d y among allv∈u+W₀^1,p(B1).

Proof Let’s prove (a). The hypothesis (iv) implies that

h→+∞lim ˆ

B₁ψ[1_AG_p(∇u_h)−1_AhG_p(∇u_h)]d y=0, ∀ψ∈C_c^∞(B₁). (5.2) Let u˜h := (1 − ψ)uh + ψu, ψ ∈ C_c^∞(B1), with 0 ≤ ψ ≤ 1. Then

∇ ˜uh=(u−uh)∇ψ+(1−ψ)∇uh+ψ∇u. Testing(Ah,u˜h), we have ˆ

B1

[Fp(∇uh)+1A_hGp(∇uh)]d y≤ ˆ

B1

[Fp(∇ ˜uh)+1A_hGp(∇ ˜uh)]d y+ηh, (5.3) where{ηh}h∈N ⊆Ris the infinitesimal sequence in (5.1). By the convexity ofFp andGp

and Lemma2.4, it follows that ˆ

B₁[F_p(∇ ˜u_h)+1_AhG_p(∇ ˜u_h)]d y

≤ ˆ

B1

[Fp((1−ψ)∇uh+ψ∇u)+1A_hGp((1−ψ)∇uh+ψ∇u)]d y +

ˆ

B1

∇Fp((u−uh)∇ψ+(1−ψ)∇uh+ψ∇u), (u−uh)∇ψd y +

ˆ

B₁∇G_p((u−u_h)∇ψ+(1−ψ)∇u_h+ψ∇u), (u−u_h)∇ψd y

≤ ˆ

B1

[(1−ψ)Fp(∇uh)+ψF_p(∇u)+1_Ah[(1−ψ)Gp(∇uh)+ψGp(∇u)]d y +c(p,L, β)

ˆ

B

(μ²+ |(u−uh)∇ψ+(1−ψ)∇uh+ψ∇u|²)^p−12 |(u−uh)∇ψ|d y.

Using the previous one in (5.3), we obtain ˆ

B₁ψ[F_p(∇u_h)+1_AhG_p(∇u_h)]d y

≤ ˆ

B1

ψ[Fp(∇u)

+1AhGp(∇u)]d y+c(p,L, β) ˆ

B

(μ²+ |(u−uh)∇ψ +(1−ψ)∇uh+ψ∇u|²)^p−12 |(u−u_h)∇ψ|d y+ηh.

(5.4)

The second term in the right hand side is infinitesimal; indeed, using the Hölder inequality, we have

ˆ

B(μ²+ |(u−u_h)∇ψ+(1−ψ)∇u_h+ψ∇u|²)^p−12 |(u−u_h)∇ψ|d y

≤ u−u_hL^p(B1)

ˆ

B₁(μ^p+ |(u−u_h)∇ψ|^p+ |(1−ψ)∇u_h|^p+ |ψ∇u|^p)d y ^p−1

p ,

which tends to 0 ashapproaches+∞. So we can inglobe the second term in the right hand side of (5.4) inηh. Add

ˆ

B1

ψ1AGp(∇uh)d yto both sides in (5.4) in order to obtain ˆ

B₁ψ[F_p(∇u_h)+1_AG_p(∇u_h)]d y≤ ˆ

B₁ψ[F_p(∇u)+1_AhG_p(∇u)]d y +

ˆ

B1

ψ[1AGp(∇uh)−1A_hGp(∇uh)]d y+ ˜ηh, where{ ˜ηh}h∈N ⊆ Ris infinitesimal. Thanks to (5.2), we can pass to the upper limit and obtain

lim sup

h→+∞

ˆ

B1

ψ[F_p(∇u_h)+1_AG_p(∇u_h)]d y≤ ˆ

B1

ψ[F_p(∇u)+1_AG_p(∇u)]d y. Finally, by lower semicontinuity, we gain

h→+∞lim ˆ

B1

ψ[Fp(∇uh)+1AGp(∇uh)]d y= ˆ

B1

ψ[Fp(∇u)+1AGp(∇u)]d y. (5.5) By the strong quasi-convexity ofF_pandG_pand Lemma2.6, we have

ˆ

B1

ψ|V(∇uh)−V(∇u)|²d y

≤c(n,p) ˆ

B₁(μ²+ |∇u_h|²+ |∇u|²)^p−22 |∇u_h− ∇u|²d y

≤c(n,p,l) ˆ

B₁[ψ(F_p(∇u_h)−F_p(∇u))− ∇F_p(∇u), ψ(∇u_h− ∇u)]d y +

ˆ

B₁[ψ1_A(G_p(∇u_h)−G_p(∇u))−1_A∇G_p(∇u), ψ(∇u_h− ∇u)]d y

.

(5.6)

Leth→ +∞in (5.6). By theii)and (5.5), we infer

h→+∞lim ˆ

B₁ψ|V(∇u_h)−V(∇u)|²d y=0.

Thanks to Lemma2.7and the arbitrariety ofψ, we conclude thatu_h →uinW_loc^1,p(B₁). Let’s proveb). Sinceλh → +∞and the energies are bounded by an appropriate constantc, it holds that

P(A_h,B₁)≤ c λh.

Leth → +∞in the previous inequality. By semicontinuity we infer that P(A,B1) = 0.

Thanks to isoperimetric inequality it follows that A= ∅or A= B. We’ll discuss the case A= ∅, being the other one similar. Forhlarge enough, by the isoperimetric inequality we have

Lⁿ(A_h)=min{Lⁿ(A_h),Lⁿ(B₁\A_h)} ≤c(n) c λh

_n−1ⁿ .

Denoting 1h(ρ)=1A_h∩∂B_ρ, for allh∈Nandρ∈(0,1), the coarea formula provides that Lⁿ(A_h)=

ˆ ₁

0

dρ ˆ

∂B_ρ

1_h(ρ)dHⁿ⁻¹≤c(n) c λh

_n−1ⁿ , which means that the sequence of functions

λh

´

∂B_ρ1_h(ρ)dHⁿ⁻¹

h∈N

converges to 0 in L¹(0,1). Thus, it converges to 0 for almost everyρ ∈ (0,1). Then, for everyρ ∈ (0,1) fixed, we can find a sequence{ρh}h∈N⊆

ρ,^1+ρ₂

such that λh

ˆ

∂B_ρh

1h(ρh)dHⁿ⁻¹→0, (5.7)

ashapproaches+∞. Comparing{(u_h,A_h)}h∈Nand{(u_h,A_h\B_ρh)}h∈N, using (5.7) and the equality

P(Ah\B_ρh,B1)=P(Ah,B1\B_ρh)+ ˆ

∂Bρh

1h(ρh)dHⁿ⁻¹, there exists an infinitesimal sequence{ηh}h∈N⊆Rsuch that

λhP(A_h,B_ρh)≤λhP(A_h,B₁)≤λhP(A_h\B_ρh,B₁)+ηh

=λhP(Ah,B1\B_ρh)+λh

ˆ

∂Bρh

1h(ρh)dHⁿ⁻¹+ηh

=λh

ˆ

∂B_ρh

1_h(ρh)dHⁿ⁻¹+ηh, providedhis so large thatA_h⊆Bρ+1

2

. Thus, thanks to (5.7) the sequence{λhP(A_h,B_ρh)}h∈N

is infinitesimal and we can conclude that

λhP(A_h,B_ρ)→0, ashapproaches+∞, sinceρh > ρ.

Let’s provec). Comparing(Ah,uh)with(Ah,u˜h)=(Ah,uh+ϕ), whereϕ ∈C¹(B1) and supp(ϕ)⊆B_ρ, we have

ˆ

B_ρ

[Fp(∇uh)+1AhGp(∇uh)]d y≤ ˆ

B_ρ

[Fp(∇ ˜uh)+1AhGp(∇ ˜uh)]d y+ηh,

with{ηh}h∈N ⊆ Rinfinitesimal and ρ ∈ (0,1) arbitrary. Thanks toa), we can use the dominated convergence theorem in order to pass to the limit ashapproaches+∞, obtaining

ˆ

B_ρ[F_p(∇u)+1_AG_p(∇u)]d y≤ ˆ

B_ρ[F_p(∇(u+ϕ))+1_AG_p(∇(u+ϕ))]d y.

By the arbitrariety ofρandϕwe can conclude the proof.

The following theorem is the main tool for proving Proposition5.1.

Theorem 5.4 (Energy decay estimate) Let K ⊆Ωbe a compact set,δ=dist(K, ∂Ω) >0 andε∈(0,1). Letc˜= ˜c(p,l,L, α, β, μ)and c_H =c_H(n,p,l,L, α, β)the constants of Lemma2.5and Corollary2.9for

H(w)= ˆ

B1

[Fp(∇w)+G_p(∇w)]d x.

Moreover, letτ ∈(0,1)such thatτ^ε< _2(1+ω¹

nc˜). Then there exist two positive constantsγ andθsuch that for any solution(u,A)of the problem(P)and for any ball B_ρ(y)with y∈K andρ∈(0,₂^δ)the two estimates

ˆ

B_ρ

[Fp(∇u)+1AGp(∇u)]d x+P(A,B_ρ)≤γρⁿ⁻¹, ρⁿ≤θ

ˆ

B_ρ

[F_p(∇u)+1_AG_p(∇u)]d x+P(A,B_ρ)

, imply that

ˆ

B_τρ(y)[F_p(∇u)+1_AG_p(∇u)]d x+P(A,B_τρ(y))

≤c_H (1+β)L l τ^n−ε

ˆ

B_ρ

[Fp(∇u)+1AGp(∇u)]d x+P(A,B_ρ)

.

Proof Let’s suppose by contradiction that there exist two sequences{γh}h∈Nand{θh}h∈N

which tend to 0, a sequence of minimizing couples{(wh,Dh)}h∈Nof (P) and a sequence of balls{B_ρh(xh)}_h∈N, withx_h ∈ K andρh ∈(0,^δ₂), for allh∈N, such that these estimates hold:

ˆ

B_ρh(xh)[Fp(∇wh)+1D_hGp(∇wh)]d x+P(Dh,B_ρh(xh))=γhρ_hⁿ⁻¹, (5.8) ρ_hⁿ ≤θh

ˆ

B_ρh(xh)[Fp(∇wh)+1D_hGp(∇wh)]d x+P(Dh,B_ρh(xh))

, (5.9) ˆ

B_τρh(xh)[F_p(∇wh)+1_DhG_p(∇wh)]d x+P(D_h,B_τρh(x_h))

> c_H(1+β)L l τ^n−ε

ˆ

B_ρh(xh)[Fp(∇wh)+1_DhG_p(∇wh)]d x+P(Dh,B_ρh(xh))

. (5.10) In what follows it will be important that the sequence{wh}h∈Nis locally equibounded inΩ.

It descends from Theorem4.1once we have proved that{wh}_h∈Nis bounded inW^1,p(Ω),

which holds true; indeed, by the minimality of(wh,D_h), (F1), (1.1) and Poincaré inequality it follows that

v∈u0+Wmin^1,p(Ω)F(v, Ω)≥F(wh,Dh)≥l ˆ

Ω|∇wh|^pd x+ ˆ

Ωγd x−k ˆ

Ω|wh|^pd x

≥l ˆ

Ω|∇wh|^pd x+ ˆ

Ωγd x−2^p−1k ˆ

Ω|wh−u0|^pd x−2^p−1k ˆ

Ω|u0|^pd x

≥(l−2^p−1kλ) ˆ

Ω|∇wh|^pd x+ ˆ

Ωγd x−2^p−1k ˆ

Ω|u0|^pd x,

sincek<₂p−1^l λ. Rescale the functionswh; define uh(y):= wh(xh+ρhy)−wh

ρ_h^p−1p γ_h^1p

∈W^1,p(B1), Ah:= Dh−xh

ρh , λh = 1

γh, (5.11) wherewh = ffl

B1wh(xh+ρhy)d y, for allh ∈ N. By the usual change of variablesx :=

x_h+ρhy, we have:

ˆ

B_ρh(xh)[Fp(∇wh)+1D_hGp(∇wh)]d x+P(Dh,B_ρh(xh))

=γhρ_hⁿ⁻¹ ˆ

B1

[Fp(∇uh)+1AhGp(∇uh)]d y+λhP(Ah,B1)

. Rescale the estimates (5.8), (5.9) and (5.10), obtaining

ˆ

B1

[F_p(∇u_h)+1_AhG_p(∇u_h)]d y+λhP(A_h,B₁)=1, (5.12) ρh ≤θhγh, (5.13) ˆ

B_τ[F_p(∇u_h)+1_AhG_p(∇u_h)]d y+λhP(A_h,B_τ) > c_H(1+β)L

l τ^n−ε. (5.14) We want to apply Theorem5.3to the sequence{(uh,Ah)}n∈N.

Firstly, let’s prove that{(u_h,A_h)}n∈Nisλh-asymptotically minimizing. LetK⊆B₁be a compact set and{(u_h,A_h)}h∈Nsuch that{u_h}h∈Nis a bounded sequence inW^1,p(B₁)with spt(u_h−u_h)⊆KandA_h ⊆B₁with A_hA_h ⊆K.

Rescale the functionsu_h: w_h(x):=ρ_h^p−1p γ_h^1pu_h x−xh

ρh

+wh ∈W^1,p(B_ρh(xh)), D_h=xh+ρhA_h. Compare the two sequences {(wh,Dh)}h∈N and {(w_h,D_h)}h∈N: by the minimality of {(wh,Dh)}h∈Nand by (1.2) we have

ˆ

B1

[Fp(∇u_h)+1_A

hGp(∇u_h)]d y+λhP(A_h,B)

= 1 γhρ_hⁿ⁻¹

ˆ

B_ρh(xh)[Fp(∇w_h)+1_D

hGp(∇w_h)]d x+P(D_h,B_ρh(xh))

≥ 1 γhρ_hⁿ⁻¹

ˆ

B_ρh(xh)[F(∇wh)+1_DhG(∇wh)]d x+P(D_h,B_ρh(x_h))